59 research outputs found
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
A note on the optimal convergence rate of descent methods with fixed step sizes for smooth strongly convex functions
Based on a recent result by de Klerk, Glineur, and Taylor (SIAM J. Optim.,
30(3):2053--2082, 2020) on the attainable convergence rate of gradient descent
for smooth and strongly convex functions in terms of function values, a
convergence analysis for general descent methods with fixed step sizes is
presented. It covers variable metric methods as well as gradient related search
directions under angle and scaling conditions. An application to inexact
gradient methods is also presented.Comment: corrected typ
Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Riemannian Steepest Descent
We study the convergence of the Riemannian steepest descent algorithm on the
Grassmann manifold for minimizing the block version of the Rayleigh quotient of
a symmetric and positive semi-definite matrix. Even though this problem is
non-convex in the Euclidean sense and only very locally convex in the
Riemannian sense, we discover a structure for this problem that is similar to
geodesic strong convexity, namely, weak-strong convexity. This allows us to
apply similar arguments from convex optimization when studying the convergence
of the steepest descent algorithm but with initialization conditions that do
not depend on the eigengap . When , we prove exponential
convergence rates, while otherwise the convergence is algebraic. Additionally,
we prove that this problem is geodesically convex in a neighbourhood of the
global minimizer of radius
Unifying time evolution and optimization with matrix product states
We show that the time-dependent variational principle provides a unifying
framework for time-evolution methods and optimisation methods in the context of
matrix product states. In particular, we introduce a new integration scheme for
studying time-evolution, which can cope with arbitrary Hamiltonians, including
those with long-range interactions. Rather than a Suzuki-Trotter splitting of
the Hamiltonian, which is the idea behind the adaptive time-dependent density
matrix renormalization group method or time-evolving block decimation, our
method is based on splitting the projector onto the matrix product state
tangent space as it appears in the Dirac-Frenkel time-dependent variational
principle. We discuss how the resulting algorithm resembles the density matrix
renormalization group (DMRG) algorithm for finding ground states so closely
that it can be implemented by changing just a few lines of code and it inherits
the same stability and efficiency. In particular, our method is compatible with
any Hamiltonian for which DMRG can be implemented efficiently and DMRG is
obtained as a special case of imaginary time evolution with infinite time step.Comment: 5 pages + 5 pages supplementary material (6 figures) (updated
example, small corrections
Gradient-type subspace iteration methods for the symmetric eigenvalue problem
This paper explores variants of the subspace iteration algorithm for
computing approximate invariant subspaces. The standard subspace iteration
approach is revisited and new variants that exploit gradient-type techniques
combined with a Grassmann manifold viewpoint are developed. A gradient method
as well as a conjugate gradient technique are described.
Convergence of the gradient-based algorithm is analyzed and a few numerical
experiments are reported, indicating that the proposed algorithms are sometimes
superior to a standard Chebyshev-based subspace iteration when compared in
terms of number of matrix vector products, but do not require estimating
optimal parameters. An important contribution of this paper to achieve this
good performance is the accurate and efficient implementation of an exact line
search. In addition, new convergence proofs are presented for the
non-accelerated gradient method that includes a locally exponential convergence
if started in a neighbourhood of the dominant
subspace with spectral gap .Comment: 29 page
Randomized sketching of nonlinear eigenvalue problems
Rational approximation is a powerful tool to obtain accurate surrogates for
nonlinear functions that are easy to evaluate and linearize. The interpolatory
adaptive Antoulas--Anderson (AAA) method is one approach to construct such
approximants numerically. For large-scale vector- and matrix-valued functions,
however, the direct application of the set-valued variant of AAA becomes
inefficient. We propose and analyze a new sketching approach for such functions
called sketchAAA that, with high probability, leads to much better approximants
than previously suggested approaches while retaining efficiency. The sketching
approach works in a black-box fashion where only evaluations of the nonlinear
function at sampling points are needed. Numerical tests with nonlinear
eigenvalue problems illustrate the efficacy of our approach, with speedups
above 200 for sampling large-scale black-box functions without sacrificing on
accuracy.Comment: 15 page
Streaming Tensor Train Approximation
Tensor trains are a versatile tool to compress and work with high-dimensional
data and functions. In this work we introduce the Streaming Tensor Train
Approximation (STTA), a new class of algorithms for approximating a given
tensor in the tensor train format. STTA accesses
exclusively via two-sided random sketches of the original data, making it
streamable and easy to implement in parallel -- unlike existing deterministic
and randomized tensor train approximations. This property also allows STTA to
conveniently leverage structure in , such as sparsity and various
low-rank tensor formats, as well as linear combinations thereof. When Gaussian
random matrices are used for sketching, STTA is admissible to an analysis that
builds and extends upon existing results on the generalized Nystr\"om
approximation for matrices. Our results show that STTA can be expected to
attain a nearly optimal approximation error if the sizes of the sketches are
suitably chosen. A range of numerical experiments illustrates the performance
of STTA compared to existing deterministic and randomized approaches.Comment: 21 pages, code available at https://github.com/RikVoorhaar/tt-sketc
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